Corollary REF  should be contrasted with the classical result that for every system \(\mathbf {X} = (X,\mu ,T)\) , every \(D\subseteq X\) , and all \(c<\mu (D)^{2}\) , \(R_{c}(D)\)  contains a set of the form \(U\setminus C\) , where \(U\subseteq \mathbb {Z}\)  is a Bohr neighborhood of 0 and \(d^{*}(C)=0\) . Furthermore, if \(\mathbf {X}\)  is weak mixing, the complement \(E:=\mathbb {Z} \setminus R_{c}(D)\)  has \(d^{*}(E)=0\) . These results can be understood in terms of Fourier transforms of measures; see Section 1.3 of [1]} for a brief exposition.
